Let r (K p , q , t) be the maximum number of colors in an edge-coloring of the complete bipartite graph K p , q not having t edge-disjoint rainbow spanning trees. We prove that r (K p , p , 1) = p 2 - 2 p + 2 for p ≥ 4 and r (K p , q , 1) = p q - 2 q + 1 for p > q ≥ 4 . Let t ≥ 2 . We also show that r (K p , p , t) = p 2 - 2 p + t + 1 for p ≥ 2 t + 3 t - 3 + 4 and r (K p , q , t) = p q - 2 q + t for p > q ≥ 2 t + 3 t - 2 + 4 . [ABSTRACT FROM AUTHOR]